A theory is presented for the mechanics of the left ventricle. A linear continuum description of the myocardium is developed, which incorporates anisotropic elastic effects due to the fiber direction field. The relation between fiber tension and fiber strain contains a time-dependent activation function that drives the ventricle around its cycle. The theory is applied to simplified geometry consisting of a thick-walled finite cylinder in which fibers spiral on helical paths and terminate on planar end surfaces. The helix pitch angle varies continuously through the wall. The ventricular cycle is analyzed by specifying the pressures at which the aortic and mitral valves open and close. Key quantities are tabulated which permit a simple determination of the properties of the model under changes of wall thickness, fiber angles, muscle parameters, preload, afterload, etc. It is shown how the active muscle parameters can be inferred from a measurement of the end systolic pressure-volume line.